Semicontinuity and continuous selections for the multivalued superposition operator without assuming growth-type conditions

Hông Thái Nguyêñ¹

¹ Institute of Mathematics Szczecin University Wielkopolska 15 70-451 Szczecin, Poland

Studia Mathematica, Tome 163 (2004) no. 1, pp. 1-19

Résumé

Let ${\mit \Omega }$ be a measure space, and $E$, $F$ be separable Banach spaces. Given a multifunction $f:{\mit \Omega }\times E\to 2^F$, denote by $N_f(x)$ the set of all measurable selections of the multifunction $f(\cdot ,x(\cdot )): {\mit \Omega }\to 2^{F}$, $s\mapsto f(s,x(s))$, for a function $x: {\mit \Omega }\to E$. First, we obtain new theorems on $H$-upper/$H$-lower//lower semicontinuity (without assuming any conditions on the growth of the generating multifunction $f(s,u)$ with respect to $u$) for the multivalued (Nemytski{ĭ}) superposition operator $N_f$ mapping some open domain $G\subset X$ into $2^Y$, where $X$ and $Y$ are Köthe–Bochner spaces (including Orlicz–Bochner spaces) of functions taking values in Banach spaces $E$ and $F$ respectively. Second, we obtain a new theorem on the existence of continuous selections for $N_f$ taking nonconvex values in non-$L_p$-type spaces. Third, applying this selection theorem, we establish a new existence result for the Dirichlet elliptic inclusion in Orlicz spaces involving a vector Laplacian and a lower semicontinuous nonconvex-valued right-hand side, subject to Dirichlet boundary conditions on a domain ${\mit \Omega }\subset {\mathbb R}^2$.

Détail

DOI : 10.4064/sm163-1-1

Keywords: mit omega measure space separable banach spaces given multifunction mit omega times denote set measurable selections multifunction cdot cdot mit omega mapsto s function mit omega first obtain theorems h upper h lower lower semicontinuity without assuming conditions growth generating multifunction respect multivalued nemytski superposition operator mapping domain subset where bochner spaces including orlicz bochner spaces functions taking values banach spaces respectively second obtain theorem existence continuous selections taking nonconvex values non l p type spaces third applying selection theorem establish existence result dirichlet elliptic inclusion orlicz spaces involving vector laplacian lower semicontinuous nonconvex valued right hand side subject dirichlet boundary conditions domain mit omega subset mathbb